Chaotic behavior of the $P$-adic Potts-Bethe mapping

In our previous investigations, we have developed the renormalization group method to $p$-adic models on Cayley trees, this method is closely related to the investigation of $p$-adic dynamical systems associated with a given model. In this paper, we study chaotic behavior of the Potts-Bethe mapping. We point out that a similar kind of result is not known in the case of real numbers (with rigorous proofs).


introduction
In [7] it was studied the thermodynamic behavior of the central site of an Ising spin system with ferromagnetic nearest-neighbor interactions on a Cayley tree by recursive methods. This method opened new perspectives between the recursion approach and the theory of dynamical systems. The existence of the phase transition is closely connected to the existence of chaotic behavior of the associated dynamical system which is governed by the Potts-Bethe function. Investigation the dynamics of this function has been the object of no small amount of study in the real and complex settings. This deceptively simple family of rational functions has given rise to a surprising number of interesting dynamical features (see for example [1,6,8,13,21]).
On the other hand, recently, polynomials and rational maps of p-adic numbers Q p have been studied as dynamical systems over this field [4,5]. It turns out that these p-adic dynamical systems are quite different to the dynamical systems in Euclidean spaces (see for example, [2,9,10,18,20,35] and their bibliographies therein). In theoretical physics, the interest in p-adic dynamical systems was started with the development of p-adic models [18,23,26]. In these investigations, it was stresses the importance of the detecting chaos in the p-adic setting [16,36,38]. In [24] it has been developed the renormalization group method to study phase transitions for several p-adic models on Cayley trees. In [22,23,25,30] we studied q-state p-adic Potts model on Cayley tree of order k (or Bethe lattice). To investigate the phase transition for this model, we need to investigated dynamical behavior of the Potts-Bethe mapping (see Appendix) (1.1) Most of the results have been obtained in case k = 2 and q is devisable by p, i.e. |q| p < 1 (see [30]). Namely, in [26] we have prove that the existence of the phase transition for the model implies the chaoticity of the Potts-Bethe mapping (in the mentioned regime). For the case k = 3 and p ≡ 2(mod3), the stability of the mapping (1.1) have been studied in [34]. Note that case k = 1 it has been investigated in [11,29].
In the present paper, we study dynamics of the p-adic Potts-Bethe mapping in a general setting. Namely, we consider k is arbitrary and q is not divisible p (The case when |q| p < 1 will be considered elsewhere). The main result of the present paper is the following one. We point out that the necessary and sufficient conditions for the existence of k √ 1 − q in Q p are given in [28]. If we take q = 2 in (1.1) then the mapping is reduced to the Ising mapping, so from 1 Theorem 1.1 and [28, Theorem 6.1] we immediately get the following corollary for the Ising mapping which covers a result of [27].
Corollary 1.2. Let p ≥ 3 and p−1 (m,p−1) is even (where k = mp s , s ≥ 0 and (m, p − 1) stands for common divisor of m and p − 1). Assume that X be a set defined as (3.3). If |k| p > |θ − 1| p then the Ising mapping f θ over the Julia set is conjugate to the full shift (Σ, σ).
In the real numbers case, analogous results with rigorous proofs are not known in the literature. In this direction, only numerical analysis predict the existence of chaos (see for example, [1,6,21]). The advantage of the non-Archimedeanity of the norm allowed us rigorously to prove the existence of the chaos (in Devaney's sense).
We note that the dynamical properties of the fixed points of some p-adic rational maps have been studied in [3-5, 12, 14, 31]. However, the global dynamical structure of rational maps on Q p remains unclear. Note that some p-adic chaotic dynamical systems have been studied in [10,38].
Note that, in the p-adic setting, due to lack of the convex structure of the set of p-adic Gibbs measures, it was quite difficult to constitute a phase transition with some features of the set of p-adic Gibbs measures. The main main Theorem 1.1 yields that the set of p-adic Gibbs measures is huge. Moreover, the advantage of the present work allows to find lots of periodic Gibbs measures.

Preliminaries
2.1. p-adic numbers. Let Q be the field of rational numbers. For a fixed prime number p, every rational number x = 0 can be represented in the form x = p r n m , where r, n ∈ Z, m is a positive integer, and n and m are relatively prime with p: (p, n) = 1, (p, m) = 1. The p-adic norm of x is given by This norm is non-Archimedean and satisfies the so called strong triangle inequality |x + y| p ≤ max{|x| p , |y| p }.
We recall a nice property of the norm, i.e. if |x| p > |y| p then |x ± y| p = |x| p . Note that this is a crucial property which is proper to the non-Archimedenity of the norm. The completion of Q with respect to the p-adic norm defines the p-adic field Q p . Any p-adic number x = 0 can be uniquely represented in the canonical form where γ(x) ∈ Z and the integers x j satisfy: [19,37]). In this case For each a ∈ Q p , r > 0 we denote Elements of the set Z p = {x ∈ Q p : |x| p ≤ 1} are called p-adic integers.
The p-adic exponential is defined by which converges for every x ∈ B p −1/(p−1) (0). Denote This set is the range of the p-adic exponential function. It is known [19] the following fact.
Lemma 2.2. The set E p has the following properties: (a) E p is a group under multiplication; Lemma 2.3. Let k ≥ 2 and p ≥ 3. Then for any α, β ∈ E p there exists γ ∈ E p such that Proof. Let p ≥ 3. Take any α, β ∈ E p . If α = β then we get According to Lemma 2.2 we get α k−1 ∈ E p . So, in this case we may take γ = α k−1 . Now, we assume that α = β. Then according to Lemma 2.2 it holds α β ∈ E p . For convenience we denote δ = α β . We show that p 1−n < |n!| p for any n ≥ 2. Indeed, it is clear that On the other hand, we have The last one with (2.4) yields The equality Choosing γ ∈ E p as follows . This completes the proof.
can be written as a finite disjoint union of balls of centers a j and of the same radius r such that for each j ∈ I there is an integer τ j ∈ Z such that For such a map f , define its Julia set by It is clear that For any i ∈ I, we let (the second equality holds because of the expansiveness and of the ultrametric property). Then define a matrix A = (a ij ) I×I , called incidence matrix as follows Here the irreducibility of A means, for any pair ij is the entry of the matrix A m . Given I and the irreducible incidence matrix A as above. Denote which is the corresponding subshift space, and let σ be the shift transformation on Σ A . We equip Σ A with a metric d f depending on the dynamics which is defined as follows.
It is clear that d f defines the same topology as the classical metric which is defined by d(x, y) = p −n(x,y) .
Theorem 2.5. [10] Let (X, J f , f ) be a transitive p-adic weak repeller with incidence matrix A. Then the dynamics (J f , f, | · | p ) is isometrically conjugate to the shift dynamics (Σ A , σ, d f ).

The fixed points of the Potts-Bethe function
In this section, for a given a integer q ≥ 2 we study fixed points of the Potts-Bethe mapping In what follows we assume that p ≥ 3 and |q| p = 1. Let x (0) be a fixed point of an analytic function f (x) and The fixed point x (0) is called attractive if 0 < |λ| p < 1, indifferent if |λ| p , and repelling if |λ| p > 1 (see [2]). Let x (0) be an attractive fixed point of the function f . We define basin of attraction of this point as follows Proof. Let |q| p = 1. Pick any x ∈ E p . According to Lemma 2.2 we get Using non-Archimedean norm's property one has Noting |k| p ≤ 1 and |θ − 1| p < 1 from (3.2) we obtain which is equivalent to the contractivity of f θ (In the real case it is not true. But, in the p-adic case, a function f : X → X for some compact subset X of Q p be a contraction if and only if |f (x) − f (y)| p < |x − y| p for any x, y ∈ X).
Proof. It is clear that x 0 = 1 is a fixed point for f θ . One has It means that x 0 is attracting. Let us show that A(x 0 ) = Dom(f θ ). Indeed, due to Sol p (x k + q − 1) = ∅ from Lemma 3.3 one finds |y k + q − 1| p ≥ |q − 1| p , for any y ∈ Q p .
In particularly, for any x ∈ Dom(f θ ) one has |f θ ( q,θ . According to Lemma 3.2 one gets f 2 θ (x (0) ) ∈ E p . Finally, the contractivity of f θ on E p and x 0 ∈ E p yield that that f n (x) → x 0 as n → ∞. The arbitrariness of x means that A(x 0 ) = Dom(f θ ).
Let us consider the case Sol p (x k + q − 1) = ∅. For a given q ≥ 2 with |q − 1| p = p −s , s ≥ 0 we define the set which is a finite union of disjoint balls. Here, r = p Case. s = 0 where η i be a solution of for a given ξ i ∈ Sol p (x k + q − 1), i = 1, κ p .
Case. s > 0 Then, there exists a p-adic integer η such that

Now consider two cases with respect to s.
Case. s = 0. Suppose that η(ξ −1)+ξ +q −1 ≡ 0(mod p) for any solution ξ of x k +q −1 ≡ 0(mod p). Then we get The last one with |θ − 1| p < 1 implies that Case. s > 0. Assume that |η| p = p − s k . Then by the non-Archimedean norm's property one gets |1 − q + ηθ| p = max{|1 − q| p , |η| p } Noting |η + 1| p = 1 we have q,θ . Let |η| p = p − s k . Then there exists a p-adic integer |ξ| p = 1 such that η = p s k ξ. One has Then according to Lemma 2.2 we obtain  q,θ ∪ X. On the other hand, according to Lemmas 3.1 and 3.2 one has A(x 0 ) ⊃ B (1) q,θ . So, we conclude that A(x 0 ) ⊃ Dom(f θ ) \ X. Proposition 3.6. Let p ≥ 3 and |θ − 1| p < |q − 1| p = p −s , s ≥ 0. If Sol p (x k + q − 1) = ∅. Then one has Proof. For any pair (x, y) ∈ Q 2 p we have where Then there exists an α z ∈ pZ p such that It follows from q ≡ 0(mod p) and the definition of η i that |η i + 1| p = |η i + 1 − q| p = 1. So, there exist p-adic integers β z , γ z ∈ pZ p such that It follows that |R(z)| p = |Q(z)| p = |θ − 1| p . Plugging them into (3.7) and according to Lemma 2.2 one finds Assume that x i = 1−q. Then for any z ∈ B r (x i ) there exists an α z ∈ pZ p such that z = 1−q+α z (θ−1). In this case, we get According to Lemma 2.2 from (3.7) one has Case. s > 0. Then for z ∈ B r (x i ) we can find Putting these into (3.7) and using the non-Archimedean norm's property one gets This completes the proof.

Proof of Theorem 1.1
In this section, we assume that |q − 1| p = p −s , s ≥ 0 and κ p ≥ 2. Proof. According to Proposition 3.6 it is enough to show that f −1 θ (X) ⊂ X. Let us assume that k √ 1 − q ∈ Q p . A function f θ has the following inverse branches on X: Case. s = 0. Take any x ∈ X. Then we get One can see that Plugging these into (4.1) and using the strong triangle inequality we obtain |g θ,i − x i | p < |θ − 1| p which implies that g θ,i (x) ∈ B r (x i ). Hence, f −1 θ (X) ⊂ X. Case. s > 0. Take any x ∈ X. Then, we have Putting the last ones into (4.2) one finds From this Proposition we immediately have the following Proof of Theorem 1.1. We have shown that under conditions k √ 1 − q ∈ Q p , κ p ≥ 2 and |k| p > p s(k−1) k |θ − 1| p the (X, J f θ , f θ ) triple is a p-adic repeller. Moreover, under these conditions incidence matrix A for the function f θ : X → Q p has the following form (it follows from Corollary 4.2) This means that a triple (X, J f θ , f θ ) is a transitive, hence according to Theorem 2.5 we conclude that the dynamics (J f θ , f θ , | · | p ) is isometrically conjugate to the shift dynamics (Σ A , σ, d f ).
Appendix A. p-adic measure Let (X, B) be a measurable space, where B is an algebra of subsets X. A function µ : B → Q p is said to be a p-adic measure if for any A 1 , . . . , A n ⊂ B such that A i ∩ A j = ∅ (i = j) the equality holds µ n j=1 A j = n j=1 µ(A j ).
A p-adic measure is called a probability measure if µ(X) = 1. A p-adic probability measure µ is called bounded if sup{|µ(A)| p : A ∈ B} < ∞. For more detail information about p-adic measures we refer to [15], [18].

Appendix B. Cayley tree
Let Γ k + = (V, L) be a semi-infinite Cayley tree of order k ≥ 1 with the root x 0 (whose each vertex has exactly k+1 edges, except for the root x 0 , which has k edges). Here V is the set of vertices and L is the set of edges. The vertices x and y are called nearest neighbors and they are denoted by l =< x, y > if there exists an edge connecting them. A collection of the pairs < x, x 1 >, . . . , < x d−1 , y > is called a path from the point x to the point y. The distance d(x, y), x, y ∈ V , on the Cayley tree, is the length of the shortest path from x to y.
The set of direct successors of x is defined by Observe that any vertex x = x 0 has k direct successors and x 0 has k + 1.

Appendix C. p-adic quasi Gibbs measure
In this section we recall the definition of p-adic quasi Gibbs measure (see [22]). Let Φ = {1, 2, · · · , q}, here q ≥ 2, (Φ is called a state space) and is assigned to the vertices of the tree Γ k + = (V, Λ). A configuration σ on V is then defined as a function x ∈ V → σ(x) ∈ Φ; in a similar manner one defines configurations σ n and ω on V n and W n , respectively. The set of all configurations on V (resp. V n , W n ) coincides with Ω = Φ V (resp. Ω Vn = Φ Vn , Ω Wn = Φ Wn ). One can see that Ω Vn = Ω V n−1 × Ω Wn . Using this, for given configurations σ n−1 ∈ Ω V n−1 and ω ∈ Ω Wn we define their concatenations by It is clear that σ n−1 ∨ ω ∈ Ω Vn . The (formal) Hamiltonian of p-adic Potts model is where J ∈ B(0, p −1/(p−1) ) is a coupling constant, and δ ij is the Kroneker's symbol. A construct of a generalized p-adic quasi Gibbs measure corresponding to the model is given below.
Given n ∈ N, we consider a p-adic probability measure µ (n) h,ρ on Ω Vn defined by Here, σ ∈ Ω Vn , and Z (h) n is the corresponding normalizing factor In this paper, we are interested in a construction of an infinite volume distribution with given finite-dimensional distributions. More exactly, we would like to find a p-adic probability measure µ on Ω which is compatible with given ones µ (n) h , i.e. µ(σ ∈ Ω : σ| Vn = σ n ) = µ (n) h (σ n ), for all σ n ∈ Ω Vn , n ∈ N.
(C. 4) We say that the p-adic probability distributions (C.2) are compatible if for all n ≥ 1 and σ ∈ Φ V n−1 : This condition according to the Kolmogorov extension theorem (see [17]) implies the existence of a unique p-adic measure µ h defined on Ω with a required condition (C.4). Such a measure µ h is said to be a p-adic quasi Gibbs measure corresponding to the model [22,23]. If one has h x ∈ E p for all x ∈ V \ {x (0) }, then the corresponding measure µ h is called p-adic Gibbs measure (see [30]). By QG(H) we denote the set of all p-adic quasi Gibbs measures associated with functions h = {h x , x ∈ V }. If there are at least two distinct generalized p-adic quasi Gibbs measures such that at least one of them is unbounded, then we say that a phase transition occurs.
The following statement describes conditions on h x guaranteeing compatibility of µ here and below a vectorĥ = (ĥ 1 , . . . ,ĥ q−1 ) ∈ Q q−1 p is defined by a vector h = (h 1 , h 1 , . . . , h q ) ∈ Q q p as followsĥ i = h i h q , i = 1, 2, . . . , q − 1 (C.7) and mapping F : Q q−1 p × Q p → Q q−1 p is defined by F(x; θ) = (F 1 (x; θ), . . . , F q−1 (x; θ)) with In [26] to establish the phase transition, we considered translation-invariant (i.e. h = {h x } x∈V \{x 0 } such that h x = h y for all x, y) solutions of (C.6). Then the equation (C.6) reduced to the following one h = f θ (h), (C.9) where (C.10) Hence, to establish the existence of the phase transition, when k = 2, we showed [30] that (C.9) has three nontrivial solutions if q is divisible by p. Note that full description of all solutions of the last equation has been carried out in [32] if k = 2 and in [33] if k = 3.